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NUMERICAL RANGE AND POSITIVE BLOCK MATRICES

Published online by Cambridge University Press:  11 June 2020

JEAN-CHRISTOPHE BOURIN
Affiliation:
Laboratoire de Mathématiques de Besançon, UMR 6623, CNRS, Université de Bourgogne Franche-Comté, Besançon, France email [email protected]
EUN-YOUNG LEE*
Affiliation:
Department of Mathematics, KNU-Center for Nonlinear Dynamics, Kyungpook National University, Daegu 702-701, Korea email [email protected]

Abstract

We obtain several norm and eigenvalue inequalities for positive matrices partitioned into four blocks. The results involve the numerical range $W(X)$ of the off-diagonal block $X$, especially the distance $d$ from $0$ to $W(X)$. A special consequence is an estimate,

$$\begin{eqnarray}\text{diam}\,W\left(\left[\begin{array}{@{}cc@{}}A & X\\ X^{\ast } & B\end{array}\right]\right)-\text{diam}\,W\biggl(\frac{A+B}{2}\biggr)\geq 2d,\end{eqnarray}$$
between the diameters of the numerical ranges for the full matrix and its partial trace.

Type
Research Article
Copyright
© 2020 Australian Mathematical Publishing Association Inc.

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Footnotes

The first author was funded by the ANR Project No. ANR-19-CE40-0002 and by the French Investissements d’Avenir program, project ISITE-BFC (contract ANR-15-IDEX-03). The second author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2018R1D1A3B07043682).

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